Av(1234, 1324, 1342, 1423)
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Generating Function
\(\displaystyle \frac{x^{2}-x +1-\sqrt{x^{4}-2 x^{3}+7 x^{2}-6 x +1}}{2 x}\)
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Counting Sequence
1, 1, 2, 6, 20, 70, 254, 948, 3618, 14058, 55432, 221262, 892346, 3630680, 14885042, ...
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Implicit Equation for the Generating Function
\(\displaystyle -x F \left(x \right)^{2}+\left(x^{2}-x +1\right) F \! \left(x \right)+x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{\left(n -1\right) a \! \left(n \right)}{5+n}+\frac{\left(1+2 n \right) a \! \left(1+n \right)}{5+n}-\frac{7 \left(n +2\right) a \! \left(n +2\right)}{5+n}+\frac{3 \left(2 n +7\right) a \! \left(n +3\right)}{5+n}, \quad n \geq 4\)
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This specification was found using the strategy pack "Insertion Point Placements Tracked Fusion" and has 53 rules.

Found on July 23, 2021.

Finding the specification took 10 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{13}\! \left(x \right) &= 0\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{6}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{24}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{21}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x , 1\right)\\ F_{27}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{36}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{37}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{6}\! \left(x \right)\\ F_{33}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{36}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= y x\\ F_{37}\! \left(x , y\right) &= F_{19}\! \left(x \right)+F_{38}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{21}\! \left(x \right) F_{40}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)+F_{46}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)+F_{45}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{44}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{22}\! \left(x \right)+F_{42}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= -\frac{y \left(F_{28}\! \left(x , 1\right)-F_{28}\! \left(x , y\right)\right)}{-1+y}\\ F_{46}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)+F_{49}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{34}\! \left(x , y\right) F_{6}\! \left(x \right)\\ F_{49}\! \left(x , y\right) &= -\frac{y \left(F_{38}\! \left(x , 1\right)-F_{38}\! \left(x , y\right)\right)}{-1+y}\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{22}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{37}\! \left(x , 1\right)\\ \end{align*}\)